We propose and analyze a seamless extended Discontinuous Galerkin (DG) discretization of hyperbolic-parabolic equations on semi-infinite domains. The semi-infinite half line is split into a finite subdomain where the model uses a standard polynomial basis, and a semi-unbounded subdomain where scaled Laguerre functions are employed as basis and test functions. Numerical fluxes enable the coupling at the interface between the two subdomains in the same way as standard single domain DG interelement fluxes. A novel linear analysis on the extended DG model yields stability constraints on the finite subdomain grid size that get tighter for increasing values of the P\'eclet number. Errors due to the use of different sets of basis functions on different portions of the domain are negligible, as highlighted in numerical experiments with the linear advection-diffusion and viscous Burgers' equations. With an added damping term on the semi-infinite subdomain, the extended framework is able to efficiently simulate absorbing boundary conditions without additional conditions at the interface. A few modes in the semi-infinite subdomain are found to suffice to deal with outgoing single wave and wave train signals, thus providing an appealing model for fluid flow simulations in unbounded regions.
翻译:我们提议并分析一个无缝的扩展扩展不连续的 Galerkin (DG) 。 半无限制半线半线将半无线半线分割成一个有限的子域, 模型使用标准的多元基值, 而半无限制的子域则使用一个半无限制的子域, 缩放的 Laguerre 函数用作基础和测试功能。 数字通量使两个子域的界面能够与标准的单域 DG 互连通通通通通量相同。 对扩展的 DG 模型的新的线性分析, 使有限的子域网格尺寸产生稳定性限制, 使P\' 括号数的值增加更加紧紧。 在域的不同部分使用不同的基础函数的半无限制子域, 正如线性粘附式和粘度 Burgers 方程式的数字实验所强调的那样, 数字通量通量使两个子域的界面能够与标准的单域 DG 互连字符连接。 扩展的框架能够在有限的子域模型中有效模拟非模拟边界条件的吸收, 而增加P\'elcelbal 界面中, 提供一个单波流的 。